Fundamental limits of perfect privacy

We investigate the problem of intentionally disclosing information about a set of measurement points X (useful information), while guaranteeing that little or no information is revealed about a private variable S (private information). Given that S and X are drawn from a finite set with joint distribution pS,X, we prove that a non-trivial amount of useful information can be disclosed while not disclosing any private information if and only if the smallest principal inertia component of the joint distribution of S and X is 0. This fundamental result characterizes when useful information can be privately disclosed for any privacy metric based on statistical dependence. We derive sharp bounds for the tradeoff between disclosure of useful and private information, and provide explicit constructions of privacy-assuring mappings that achieve these bounds.